Question

Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix}2&1\\1&1\end{bmatrix}\)

Answer 1

Let A= \(\begin{bmatrix}2&1\\1&1\end{bmatrix}\)
We know that A = IA
so \(\begin{bmatrix}2&1\\1&1\end{bmatrix}\)= \(\begin{bmatrix}1&0\\0&1\end{bmatrix}A\)

\(\Rightarrow\begin{bmatrix}1&0\\1&1\end{bmatrix}=\begin{bmatrix}1&-1\\0&1\end{bmatrix}A\)      (R₁->R₁-R₂)

\(\Rightarrow\begin{bmatrix}1&0\\0&1\end{bmatrix}=\begin{bmatrix}1&-1\\-1&2\end{bmatrix}A\)   (R₂->R₂-R₁)

\(\therefore\)  A^-1= \(\begin{bmatrix}1&-1\\-1&2\end{bmatrix}\)